The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number - Paolo Orlandi (Fluids)
The mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (<semantics>Rλ</semantics>). This value is associated with the well-scaling dissipation spectrum in Kolmogorov units, where the maximum corresponds to the bottleneck peak. Even the scalar dissipation rate at the high <semantics>Rλ</semantics> considered in the present direct numerical simulations attains a constant value as <semantics>Sc</semantics> increases. In this scenario, the maximum of the scalar dissipation spectra reaches its peak within the bottleneck, starting at <semantics>Sc>0.5</semantics>. A qualitative explanation for the formation of the two bottlenecks is related to the blockage of energy transfer from large to small scales in the inertial ranges. Within the bottleneck, the self-similar, ribbon-like structures transition into the rod-like structures characteristic of the exponential decay range. Investigating the viscous dependence of the bottleneck’s amplitude may be aided by examining the evolution of a passive scalar. As <semantics>Sc</semantics> decreases, the scalar spectra undergo changes across the wave number k range. The bottleneck is dismantled, and at very low <semantics>Sc</semantics> values, the spectrum tends towards Batchelor’s theoretical prediction, diminishing proportionally to <semantics>k−17/3</semantics>. To comprehend the flow structures responsible for the bottleneck, visualizations of <semantics>θ∇2θ</semantics> and probability density functions at various <semantics>Sc</semantics> values are presented and compared with those of <semantics>ui∇2ui</semantics>. The numerical method employed for generating three-dimensional spectra and quantities such as energy and scalar variance dissipation in physical space must be accurate, particularly in resolving small scales. This paper additionally demonstrates that the second-order finite difference scheme conserving kinetic energy and scalar variance in the inviscid limit in viscous simulations accurately predicts the exponential decay range in one-dimensional and three-dimensional turbulent kinetic energy and scalar variance spectra.